时间分裂谱方法解弱非线性Klein-Gordon方程长时间动力学的一致误差界
Uniform error bounds of a time-splitting spectral method for the long-time dynamics of the nonlinear Klein-Gordon equation with weak nonlinearity
摘要：
对于具有弱幂型非线性项的Klein-Gordon方程（NKGE）和具有$O(1)$初值的非线性项，当非线性项的强度由具有常数$p \in \mathbb{N}^+$和无量纲参数$\varepsilon \in (0, 1]$的$\varepsilon^{p}$表征时，对于直到$O(\varepsilon^{-β})$的长时间动力学，利用$0 \leq β\leq p$建立了时间分裂Fourier拟谱（TSFP）方法的一致误差界.事实上，当$0 < \varepsilon \ll 1$时，问题等价于具有小初始数据和$O(1)$非线性强度的NKGE的长时间动力学，而初始数据（和解）的振幅在$O(\varepsilon)$处。通过将NKGE转化为相对论非线性薛定谔方程，我们采用TSFP方法对其进行数值离散。利用数学归纳法对数值解进行定界，证明了$h$网格尺寸、$τ$时间步长和$m\ge2$的TSFP方法在$O(h^{m}+\varepsilon^{p-β}τ^2)$处的一致误差界依赖于解的正则性.误差界对于直到$O(\varepsilon^{-β})$的时间的长时间模拟是一致准确的，并且对于$\varepsilon\in(0,1]$是一致有效的。特别地，对于参数区域$0\leβ<p$中的大时间步长$τ= O(\varepsilon^{-(p-β)/2})$，误差界在二阶速率处是一致的。数值结果证实了我们的误差范围在长时间制度。最后，将TSFP方法及其误差界推广到高振荡复NKGE，该复NKGE在空间传播波长为$O(1)$，在时间传播波长为$O(\varepsilon^β)$，在$O(\varepsilon^{-β})$传播波速.
英文摘要：
We establish uniform error bounds of time-splitting Fourier pseudospectral (TSFP) methods for the nonlinear Klein--Gordon equation (NKGE) with weak power-type nonlinearity and $O(1)$ initial data, while the nonlinearity strength is characterized by $\varepsilon^{p}$ with a constant $p \in \mathbb{N}^+$ and a dimensionless parameter $\varepsilon \in (0, 1]$, for the long-time dynamics up to the time at $O(\varepsilon^{-β})$ with $0 \leq β\leq p$. In fact, when $0 < \varepsilon \ll 1$, the problem is equivalent to the long-time dynamics of NKGE with small initial data and $O(1)$ nonlinearity strength, while the amplitude of the initial data (and the solution) is at $O(\varepsilon)$. By reformulating the NKGE into a relativistic nonlinear Schrödinger equation, we adapt the TSFP method to discretize it numerically. By using the method of mathematical induction to bound the numerical solution, we prove uniform error bounds at $O(h^{m}+\varepsilon^{p-β}τ^2)$ of the TSFP method with $h$ mesh size, $τ$ time step and $m\ge2$ depending on the regularity of the solution. The error bounds are uniformly accurate for the long-time simulation up to the time at $O(\varepsilon^{-β})$ and uniformly valid for $\varepsilon\in(0,1]$. Especially, the error bounds are uniformly at the second order rate for the large time step $τ= O(\varepsilon^{-(p-β)/2})$ in the parameter regime $0\leβ<p$. Numerical results are reported to confirm our error bounds in the long-time regime. Finally, the TSFP method and its error bounds are extended to a highly oscillatory complex NKGE which propagates waves with wavelength at $O(1)$ in space and $O(\varepsilon^β)$ in time and wave velocity at $O(\varepsilon^{-β})$.
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